STATISTICS OF VARIATIONAL DATA ASSIMILATION IN CONTINUOUS TIME

许多自然现象表现为动态过程，即随时间演变的过程。这种现象的数学描述通常由描述时间演化的方程组成，如微分方程或差分方程；我们把它们称为动态模型。大气、太阳、人类群体行为、交通和股票市场的动态模型已经被构想出来，这只是其中的几个例子。动态模型允许通过数值模拟（通常在计算机上）预测现实世界动态过程的未来行为。然而，为了预测动态过程的未来行为，必须知道它的当前状态。数据同化是这个项目的主题，意味着收集过去和现在对动态过程的观察，并以动态一致的方式估计其当前状态甚至整个轨迹。（例如，数据同化不仅会产生一系列全球风场的快照；这些快照的演变将与描述空气运动的物理一致。）因此，数据同化是动态模型预测的核心步骤。数据同化已经在广泛的应用中进行，例如在天气预报中。然而，在某种程度上，数据同化依赖于一种特殊的方法，其中只有一部分被完全理解。然而，对数据同化的透彻理解是至关重要的，因为每个预测的性能和价值都至关重要地取决于数据同化。本项目旨在为数据同化提供进一步的数学基础。特别地，以下几点将被研究：*动态模型通常是在连续时间中制定的，并且由于数学原因，在连续时间中分析数据同化比在离散时间中分析要好得多。但在实践中，天气观测是在离散时间点采样的，这似乎需要一个离散时间框架来同化数据。我们如何在保护连续时间方法的优雅和强大的同时，处理好这个重要的实际问题？*许多数据同化方法提供了看似合理的解决方案，但没有正确理解其确切性质。一个特别紧迫的问题是与估计轨迹有关的不确定性。假设观测包含测量误差，该误差将通过整个数据同化机制明显地反馈到估计的轨迹上。我们怎么处理这个？*从业者需要一种形式来对他或她的数据同化结果进行一些质量控制。这里的问题是：简单地将数据同化的输出与观察结果进行比较是危险的，因为观察结果已经被用来寻找潜在的轨迹，所以这种方法可能会给出过于乐观的结果。在统计学中，这被称为“样本评估”，并且已经设想了几种方法来避免它们。在数据同化中，也需要类似的东西，尽管由于观测结果通常高度依赖，问题更为复杂；一系列的风观测不能像医学试验中的一系列病人一样被对待。但是，在先前工作的基础上，将开发一种形式主义，允许对数据同化进行更现实的性能评估。

Many natural phenomena manifest themselves as dynamical processes, that is processes evolving in time. Mathematical descriptions of such phenomena usually consist of equations describing temporal evolution, such as differential or difference equations; we will refer to them as dynamical models. Dynamical models have been conceived for the atmosphere, the sun, human crowd behaviour, traffic, and the stock market, just to name a few. A dynamical model allows to forecast the future behaviour of the real world dynamical process through numerical simulations (usually on a computer).However, in order to forecast the future behaviour of the dynamical process, its current state has to be known. Data assimilation, which is the main theme of this project, means to gather past and present observations of the dynamical process and estimate its current state or even whole trajectories in a dynamically consistent fashion. (E.g. data assimilation will not only result in a series of snapshots of the global wind fields; the evolution of these snapshots will be consistent with the physics describing air motion.) For this reason, data assimilation is a core step in forecasting with dynamical models.Data assimilation is carried out already in a wide range of applications, for example in weather forecasting. To some extent though, data assimilation rests on an ad--hoc methodology with only part of it being completely understood. A thorough understanding of data assimilation though is vital, as the performance and thus the value of every forecast depends crucially on the data assimilation.This project aims at providing data assimilation with further mathematical foundations. In particular, the following points will be investigated:* Dynamical models are often formulated in continuous time, and data assimilation is much nicer to analyse in continuous time than in discrete time, for mathematical reasons. In practice though, weather observations are sampled at discrete points in time, which seems to necessitate a discrete time framework for data assimilation. How do we take care of this important practical problem while at the same time rescuing the elegance and power of a continuous time approach? * Many data assimilation approaches provide solutions that appear reasonable, but the precise properties are not properly understood. A particularly pressing problem is the uncertainty associated with the estimated trajectories. Suppose the observations contain measurement error, this error will clearly feed through the entire data assimilation machinery onto the estimated trajectories. How do we take care of this?* The practitioneer needs a formalism to perform some quality control of her or his data assimilation results. The problem here is this: simply comparing the output of data assimilation with the observations is dangerous, since the observations have already been used to find the underlying trajectory, so this approach might give overly optimistic results. In statistics, this is known as ``in sample evaluation'', and several methods have been conceived to avoid them. In data assimilation, something similar is needed, although the problem is more complex as the observations are usually heavily dependent; a series of wind observations cannot be treated like a series of patients in medical trials. But building on previous work, a formalism will be developed allowing for more realistic performance assessment of data assimilation.